# NUMERICAL MODELS | Mesoscale Atmospheric Modeling

## NUMERICAL MODELS | Mesoscale Atmospheric Modeling

Synopsis Mesoscale systems in the atmosphere are those in which the instantaneous pressure ﬁeld can be determined accurately by the temperature ﬁeld, but the winds, even in the absence of surface frictional effects, are out of balance with the horizontal pressure gradient force. The framework of mesoscale models is overviewed and shows that the models include a physics base core comprising the pressure gradient force, advection, and gravity, and all other physical processes that are parameterized using tuned modular representations of turbulence, longwave and shortwave radiation, cumulus, and stable cloud processes. Computational methods, lateral and initial boundary conditions, and model validation are some of the topics discussed.

Introduction Atmospheric mesoscale systems are identiﬁed as those in which the instantaneous pressure ﬁeld can be determined accurately by the temperature ﬁeld, but the winds, even in the absence of surface frictional effects, are out of balance with the horizontal pressure gradient force. The pressure ﬁeld, under this situation, is said to be ‘hydrostatic.’ Larger scale atmospheric features (which are called ‘synoptic’ weather features), in contrast, have a wind ﬁeld that is close to a balance with the horizontal pressure gradient force. These large-scale winds are said to be near gradient wind balance. Atmospheric features that are smaller than the mesoscale have pressure ﬁelds in which wind acceleration is a signiﬁcant component (which is referred to as the dynamic wind). The pressure gradient that causes this dynamic wind is called the nonhydrostatic pressure. Atmospheric mesoscale models are based on a set of conservation equation for velocity, heat, density, water, and other trace atmospheric gases and aerosols. The equation of state used in these equations is the ideal gas law. The conservation of velocity equation is derived from Newton’s second law of motion as applied to the rotating Earth. The conservation of heat equation is derived from the ﬁrst law of thermodynamics. The remaining conservation equations are written as a change in an atmospheric variable (e.g., water) in a Lagrangian framework where sources and sinks are identiﬁed. Each of these conservation equations can be written to represent the changes following a parcel of velocity, potential temperature (entropy), water in its three phases, other atmospheric gases and aerosols, and mass, including source–sink terms. Models, however, seldom express the conservation relations in a Lagrangian framework. The chain rule of calculus is used to convert to an Eulerian framework. Several assumptions are typically made in the conservation equations. These include the neglect of small-scale ﬂuctuation of density except when multiplied by gravity (this is called the Boussinesq approximation), the neglect of vertical acceleration relative to the differences between the vertical pressure gradient force and gravity (referred to as the hydrostatic assumption), and the neglect of all molecular transfers. The ﬁrst two assumptions have not been made in recent years in the models, however, since the numerical equations are actually easier to solve without these assumptions. Nonetheless, the spatial and temporal scales of mesoscale systems result

Encyclopedia of Atmospheric Sciences 2nd Edition, Volume 4

in the two assumptions being excellent approximations with respect to mesoscale-sized systems. The third assumption is justiﬁed since advection is much more signiﬁcant at transfers of heat, momentum, water, and other chemical species, than molecular motion on the mesoscale. These conservation relations that are written as a set of simultaneous, nonlinear differential equations, unfortunately, cannot be used without integrating them over deﬁned volumes of the atmosphere. These volumes are referred to as the model ‘grid volume’. The region of the atmosphere for which these grid volumes are deﬁned is called the ‘model domain.’ The integration of the conservation relations produces ‘grid volume averages,’ with point-speciﬁc values of the variables called ‘subgrid-scale values.’ The ‘resolution’ of data is limited to two grid intervals in each spatial direction. The result of the grid volume averaging produces equations for the local time derivative of the grid volume-resolved variable which includes ‘subgrid-scale ﬂuxes.’ An assumption that is routinely made in all mesoscale models (usually without additional comment) is that the gird volume average of subgrid-scale ﬂuctuations is zero. This assumption, often referred to as ‘Reynold’s averaging,’ is actually only accurate when there is a clear spatial scale separation between subgrid scale- and grid volume-resolved quantities. Mesoscale model equations have been solved in a Cartesian coordinate framework. Each coordinate in this system is perpendicular to the other two coordinates at every location. Most mesoscale models, however, transform to a generalized vertical coordinate. The most common coordinates involve some form of terrain-following transformation, where the bottom coordinate surface is terrain height or terrain surface pressure. The result of these transformations is that the new coordinate system is not orthogonal, in general. Unless this nonorthogonality is small, the correct treatment of nonhydrostatic pressure effects in mesoscale models requires the use of tensor transformation techniques, as opposed to the separate use of the chain rule on each component of velocity, separately. The use of generalized coordinate systems introduces additional sources for errors in the models, since the interpolation of variables to grid levels becomes more complicated. The model variables also need to be deﬁned on a speciﬁed grid mesh. When all dependent variables are deﬁned at the same grid points, the grid is said to be ‘nonstaggered.’ When dependent variables are deﬁned at different grid points, the grid

http://dx.doi.org/10.1016/B978-0-12-382225-3.00217-6

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Numerical Models j Mesoscale Atmospheric Modeling